Same area, different perimeter

In your book, Maths Mela (Chapter 11), Trisha makes two rectangles out of 1 cm squares. The first is 2 cm by 3 cm: it covers 6 squares, so its area is 6 square cm, and its perimeter is 2 + 3 + 2 + 3 = 10 cm. The second is 3 cm by 4 cm: area 12 square cm, perimeter 3 + 4 + 3 + 4 = 14 cm. Trisha says, 'I increased the area, and the perimeter increased.' Is that always true?

Look at some shapes that all cover 12 squares. A rectangle 6 cm by 2 cm has perimeter 6 + 2 + 6 + 2 = 16 cm. A rectangle 3 cm by 4 cm has perimeter 14 cm. A long strip 12 cm by 1 cm has perimeter 12 + 1 + 12 + 1 = 26 cm. All three have the same area, 12 square cm, but their perimeters are different. The longer and thinner the shape, the longer its border.

Now take a strip 8 cm by 1 cm. It covers only 8 squares, so its area is 8 square cm, but its perimeter is 8 + 1 + 8 + 1 = 18 cm. Compare it with the 3 cm by 4 cm rectangle: that has the BIGGER area (12 square cm) but the SMALLER perimeter (14 cm). So Trisha's idea is not always true: a bigger area does not always mean a bigger perimeter.